" name="description" /> Mathematical Proceedings of the Cambridge Philosophical Society - On the <i>L</i><sub><i>q</i></sub> norm of cyclotomic Littlewood polynomials on the unit circle - Cambridge Journals Online

Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

On the Lq norm of cyclotomic Littlewood polynomials on the unit circle

TAMÁS ERDÉLYIa1

a1 Department of Mathematics, Texas A&M University, College Station, Texas 77843, U.S.A. e-mail: terdelyi@math.tamu.edu

Abstract

Let n be the collection of all (Littlewood) polynomials of degree n with coefficients in {−1, 1}. In this paper we prove that if (P) is a sequence of cyclotomic polynomials P, then

\[
M_q(P_{2\nu}) > (2\nu +1)^a
\]

for every q > 2 with some a = a(q) > 1/2 depending only on q, where
\[
M_q(P) := \left (\frac{1}{2\pi} \int_0^{2\pi}{|P(e^{it})|^q \,dt}
\right)^{1/q}\,, \qquad q > 0\,.
\]

The case q = 4 of the above result is due to P. Borwein, Choi and Ferguson. We also prove that if (P) is a sequence of cyclotomic polynomials P, then
\[
M_q(P_{2\nu}) < (2\nu+1)^b
\]

for every 0 < q < 2 with some 0 < b = b(q) < 1/2 depending only on q. Similar results are conjectured for Littlewood polynomials of odd degree. Our main tool here is the Borwein–Choi Factorization Theorem.

(Received September 14 2010)

(Revised March 07 2011)

(Online publication July 13 2011)

Footnotes

Dedicated to the memory of Professor Paul Turán on the occasion of his 100th birthday.